What, if anything, can I do on the outside to see about getting an inmate transferred to a different facility? And is it any harder to get transferred to a facility with a work release program than to one that doesn't?

Transfers rarely happen by request, especially from someone on the outside. In federal, you have to serve 9 months in the current prison before they would even consider a transfer to another. They want a track record of excellent behavior and programming to get "stepped down" in custody levels which leads to work release.